Download ONSAGER`S VARIATIONAL PRINCIPLE AND ITS APPLICATIONS

ONSAGER’S VARIATIONAL PRINCIPLE
AND ITS APPLICATIONS
Tiezheng Qian∗
Department of Mathematics, Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong
(Dated: April 30, 2016)
Abstract
This manuscript is prepared for a Short Course on the variational principle formulated by Onsager based on his reciprocal symmetry for the kinetic coefficients in linear irreversible thermodynamic processes.
Based on the reciprocal relations for kinetic coefficients, Onsager’s variational principle is of
fundamental importance to non-equilibrium statistical physics and thermodynamics in the linear
response regime. For his discovery of the reciprocal relations, Lars Onsager was awarded the 1968
Nobel Prize in Chemistry. The purpose of this short course is to present Onsager’s variational
principle and its applications to first-year graduate students in physics and applied mathematics.
The presentation consists of four units:
1. Review of thermodynamics
2. Onsagers reciprocal symmetry for kinetic coefficients
3. Onsagers variational principle
4. Applications:
4.1 Heat transport
4.2 Lorentz reciprocal theorem
4.3 Cross coupling in rarefied gas flows
4.4 Cross coupling in a mixture of fluids
∗
Corresponding author. Email: [email protected]
1
I.
A BRIEF REVIEW OF THERMODYNAMICS
The microcanonical ensemble is of fundamental value while the canonical ensemble is
more widely used. For more details, see C. Kittel, Elementary Statistical Physics.
The energy is a constant of motion for a conservative system. If the energy of the system
is prescribed to be in the range δE at E0 , we may form a satisfactory ensemble by taking
the density as equal to zero except in the selected narrow range δE at E0 : P (E) = constant
for energy in δE at E0 and P (E) = 0 outside this range. This particular ensemble is known
as the microcanonical ensemble. It is appropriate to the discussion of an isolated system
because the energy of an isolated system is a constant.
Let us consider the implications of the microcanonical ensemble. We are given an isolated
classical system with constant energy E0 . At time t0 the system is characterized by definite
values of position and velocity for each particle in the system. The macroscopic average
physical properties of the system could be calculated by following the motion of the particles
over a reasonable interval of time. We do not consider the time average, but consider instead
an average over an ensemble of systems each at constant energy within δE at E0 . The
microcanonical ensemble is arranged with constant density in the region of phase space
accessible to the system. Here we have made as a fundamental postulate the assumption of
equal a priori probabilities for different accessible regions of equal volume in phase space.
A.
Entropy
The entropy of the microcanonical ensemble is
S = kB log ∆Γ,
where ∆Γ is the number of states with energy distributed between E0 and E0 + ∆E. Re∑
member that the general definition of the entropy is S = −kB n wn log wn , where kB
is the Boltzmann constant and wn is the distribution function for the system, satisfying
∑
n wn = 1, with n being the set of all quantum numbers which denote the various stationary states of the system. The entropy of the microcanonical ensemble is obtained from a
∑
maximization of S = −kB n wn log wn with respect to wn , subject to the normalization
∑
condition n wn = 1 and the energy condition that wn is nonzero only if the corresponding
energy is in the selected narrow range δE at E0 .
2
B.
Conditions for equilibrium
The entropy is a maximum when a closed system is in equilibrium. The value of S for
a system in equilibrium depends on the energy U of the system; on the number Ni of each
molecular species i of the system; and on external variables xν , such as volume V , strain,
magnetization. In other words,
S = S(U, xν , Ni ).
We consider the condition for equilibrium in a system made up of two interconnected subsystems, as illustrated in figure 1. Initially the subsystems are separated from each other by
a insulating, rigid, and non-permeable barrier.
Weakly interacting quasi-closed subsystems: In the following discussion, we consider two
weakly interacting quasi-closed subsystems 1 and 2. On the one hand, the two subsystems are
quasi-closed, and hence Sj = Sj (Uj , Vj , Nij ) for j = 1, 2 and S = S1 + S2 from the statistical
independence. On the other hand, weak coupling is assumed between the two subsystems. This
allows the whole (closed) system to relax towards complete equilibrium (e.g., via energy transfer
between the two subsystems) if the total entropy is not maximized (with respect to U1 or U2 ).
1.
Thermal equilibrium
Imagine that the barrier is allowed to transmit energy (beginning at one instant of time),
with other inhibitions remaining in effect. If the conditions of the two subsystems 1 and
2 do not change, then they are in thermal equilibrium and the entropy of the total system
must be a maximum with respect to small transfer of energy from one subsystem to the
other. Using the additive property of the entropy,
S = S1 + S2 ,
we have in equilibrium
δS = δS1 + δS2 = 0,
or
(
δS =
∂S1
∂U1
)
(
δU1 +
3
∂S2
∂U2
)
δU2 = 0.
Because the total system is thermally closed and the total energy is constant, i.e., δU =
δU1 + δU2 = 0, we have
[(
δS =
∂S1
∂U1
)
(
−
∂S2
∂U2
)]
δU1 = 0.
As δU1 is an arbitrary variation, we obtain in thermal equilibrium
∂S1
∂S2
=
.
∂U1
∂U2
Defining the temperature T by
1
=
T
(
∂S
∂U
)
,
xν ,Ni
we obtain T1 = T2 as the condition for thermal equilibrium.
Suppose that the two subsystems were not originally in thermal equilibrium, but that
T2 > T1 . When thermal contact is established to allow energy transmission, the total entropy
S will increase. (The removal of any constraint can only increase the volume of phase space
accessible to the system.) Thus, after thermal contact is established, δS > 0, or
[(
) (
)]
∂S1
∂S2
−
δU1 > 0,
∂U1
∂U2
and
[
]
1
1
−
δU1 > 0.
T1 T2
Assuming T2 > T1 , we have δU1 > 0. This means that energy passes from the system of
high T to the system of low T . So T is indeed a quantity that behaves qualitatively like a
temperature.
2.
Mechanical equilibrium
Now imagine that the wall is allowed to move and also transmit energy, but does not pass
particles. The volumes V1 and V2 of the two subsystems can readjust to (further) maximize
the entropy. In mechanical equilibrium
)
(
)
(
)
(
)
(
∂S2
∂S1
∂S2
∂S1
δV1 +
δV2 +
δU1 +
δU2 = 0.
δS =
∂V1
∂V2
∂U1
∂U2
After thermal equilibrium has been established, we have
)
(
)
(
∂S2
∂S1
δU1 +
δU2 = 0.
∂U1
∂U2
4
As the total volume V = V1 + V2 is constant, we have δV = δV1 + δV2 = 0, and
[(
) (
)]
∂S1
∂S2
δS =
−
δV1 = 0.
∂V1
∂V2
As δV1 is an arbitrary variation, we obtain in mechanical equilibrium
∂S1
∂S2
=
.
∂V1
∂V2
Defining the pressure Π by
Π
=
T
(
∂S
∂V
)
,
U,Ni
we see that for a system in thermal equilibrium (with T1 = T2 ), Π1 = Π2 is the condition for
mechanical equilibrium. In general we define a generalized force Xν related to the coordinate
xν by the equation
Xν
=
T
(
∂S
∂xν
)
.
U,Ni
It is interesting to note that from the defining equations for T and Π, we obtain
(
)
(∂S/∂V )U,Ni
∂U
.
Π=
=−
(∂S/∂U )V,Ni
∂V S,Ni
This expression for the pressure will be derived again by regarding U as a function of S and
V.
Suppose that the two subsystems in thermal equilibrium were not originally in mechanical
equilibrium, but that Π1 > Π2 . When the wall is allowed to move, the total entropy S
will increase. (The removal of any constraint can only increase the volume of phase space
accessible to the system.) From
) (
)]
[(
∂S1
∂S2
1
−
δV1 = [Π1 − Π2 ] δV1 > 0,
δS =
∂V1
∂V2
T
we see that Π1 > Π2 requires δV1 > 0: the subsystem of the higher pressure expands in
volume. So Π is indeed a quantity that behaves qualitatively like a pressure.
3.
Particle equilibrium
Now imagine that the wall allows diffusion through it of molecules of the ith chemical species. Suppose thermal equilibrium and mechanical equilibrium have already been
established. From δNi1 + δNi2 = 0 and
) (
)]
[(
∂S2
∂S1
−
δNi1 = 0,
δS =
∂Ni1
∂Ni2
5
we obtain
∂S1
∂S2
=
,
∂Ni1
∂Ni2
as the condition for particle equilibrium. Defining the chemical potential µi by
(
)
µi
∂S
− =
,
T
∂Ni U,xν
we see that for a system in both thermal equilibrium (T1 = T2 ) and mechanical equilibrium
(Π1 = Π2 ), µi1 = µi2 is the condition for particle equilibrium. It is easy to show that
particles tend to move from a region of higher chemical potential to that of a lower chemical
potential as the system approaches equilibrium (δS > 0).
C.
Connection between statistical and thermodynamical quantities
For a system in equilibrium, S = S(U, xν , Ni ), where U is the energy, xν denotes the set
of external parameters describing the system, and Ni is the number of molecules of the i-th
species. If the conditions are changed slightly, but reversibly in such a way that the resulting
system is also in equilibrium, we have
)
(
∑ ( ∂S )
∑ ( ∂S )
∂S
dU
1∑
1∑
dU +
dxν +
dNi =
+
Xν dxν −
µi dNi ,
dS =
∂U
∂x
∂N
T
T
T
ν
i
ν
ν
i
i
which may be rewritten as
dU = T dS −
∑
Xν dxν +
∑
ν
µi dNi .
i
Consider that the number of particles is fixed and the volume is the only external parameter:
dNi = 0; xν ≡ V ; Xν ≡ Π. Then,
dU = T dS − ΠdV.
We see that the change of internal energy consists of two parts. The term T dS represents
the change in U when the external parameters are kept constant (dV = 0). This is what we
mean by heat. Thus
DQ = T dS
is the quantity of heat added to the system in a reversible process. The symbol D is used
instead of d because DQ is not an exact differential — that is, Q is not a state function. The
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term −ΠdV is the change in internal energy caused by the change in external parameters;
this is what we mean by mechanical work, and
DW = −ΠdV
is the work done on the system through the volume change dV . By elementary mechanics
DW = −pdV , hence Π ≡ p. Therefore the change in internal energy may be expressed as
dU = DQ + DW,
which is The First Law of Thermodynamics.
That dS = DQ/T is an exact differential in a reversible process is a statement of The
Second Law of Thermodynamics. That is, DQ/T is a differential of a state function, entirely
defined by the state of the system. Now we know that state function is the entropy.
On S = S(U, V ): Here the discussion on various thermodynamic quantities starts from S =
S(U, V ) in equilibrium and dS = dU/T + pdV /T for reversible processes. The evaluation of S as a
function of U and V can be carried out using the microcanonical ensemble which is characterized
by U and V .
We are ready to express U as a function of S and V : U = U (S, V ). Other quantities of
interest are then obtained from U (S, V ):
dU = T dS − pdV =
from which
(
∂U
∂S
(
T =
and
(
)
∂U
∂S
dS +
V
∂U
∂V
)
dV,
S
)
;
V
(
)
∂U
p=−
.
∂V S
(Note that the expression for p has been obtained before by starting from S = S(U, V ).)
However, S and V are often inconvenient independent variables — it is more convenient to
work with (T, V ) or (T, p). For this purpose we introduce some thermodynamic potentials:
F and G.
1.
Helmholtz free energy
When the temperature T is set, it is convenient to introduce the Helmholtz free energy
F which is defined as F ≡ U − T S, regardless of whether the system is in equilibrium or not
7
(as long as the entropy can be defined via partial equlibria). For a system in equilibrium,
the macroscopic state is completely determined by T and V , and F may be expressed as a
function of these two independent variables describing the equilibrium state, i.e.,
F (T, V ) ≡ U − T S,
(
∂U
where T =
∂S
T dS − pdV and
)
has been used to replace S as an independent variable. From dU =
V
(
dF = dU − T dS − SdT = −SdT − pdV =
we obtain
(
−S =
and
(
−p =
∂F
∂T
∂F
∂V
∂F
∂T
)
(
dT +
V
∂F
∂V
)
dV,
T
)
,
V
)
.
T
Therefore, if the Helmholtz free energy F is expressed as a function of independent variables
T and V , then S and p are readily calculated from its partial derivatives.
On the Helmholtz free energy F : While the definition of F ≡ U − T S is applicable to
arbitrary macroscopic states (so long as there is partial equilibrium and hence the entropy can
be defined), the total differential dF = −SdT − pdV is obtained for reversible processes only.
That is, −S = (∂F /∂T )V and −p = (∂F /∂V )T are relations for equilibrium states only. For
an equilibrium system of fixed T and V , F (T, V ) can be obtained from the partition function of
canonical ensemble.
Consider a thermodynamic process at constant temperature T . The change in F is given
by ∆F = ∆U − T ∆S = (∆Q − T ∆S) + ∆W . For a reversible process, ∆Q = T ∆S, and
hence ∆F = ∆W . For an irreversible process, however, ∆Q < T ∆S according to the second
law of thermodynamics, and hence ∆F < ∆W . That −∆W ≤ −∆F means the maximum
work which can be done by the system is the decrease of F . If the volume is fixed, then
∆W = 0 and ∆F ≤ 0. That is, at constant temperature and volume, the Helmholtz free
energy varies towards a minimum, which is just F (T, V ) for the final equilibrium state.
On ∆Q ≤ T ∆S : Consider a system in contact with a large heat bath such that a constant
temperature T is maintained. If ∆Q is the heat received by the system from the heat bath, then
the heat received by the heat bath from the system is −∆Q. Assuming that the (very small)
8
change in the (very large) heat bath is reversible, we have −∆Q/T as the change of its entropy.
That the total entropy tends to increase means −∆Q/T + ∆S ≥ 0, i.e., ∆Q ≤ T ∆S.
2.
Gibbs free energy
When the temperature T and the pressure p are both set, it is convenient to introduce
the Gibbs free energy G which is defined as G ≡ F + pV ≡ U − T S + pV , regardless of
whether the system is in equilibrium or not (as long as the entropy can be defined via partial
equlibria). For a system in equilibrium, the macroscopic state is completely determined by
T and p, and G may be expressed as a function of these two independent variables describing
the equilibrium state, i.e.,
G(T, p) ≡ U − T S + pV,
(
)
∂U
∂U
where T =
and p = −
has been used to replace S and V as the two
∂S V
∂V S
independent variables. From dU = T dS − pdV and
(
(
)
)
∂G
∂G
dT +
dp,
dG = dU − T dS − SdT + pdV + V dp = −SdT + V dp =
∂T p
∂p T
(
)
we obtain
(
−S =
and
(
V =
∂G
∂T
∂G
∂p
)
,
p
)
.
T
Therefore, if the Gibbs free energy G is expressed as a function of independent variables T
and p, then S and V are readily calculated from its partial derivatives.
Consider a thermodynamic process at constant temperature T and pressure p. The change
in G is given by ∆G = ∆U − T ∆S + p∆V = (∆Q − T ∆S) + ∆W + p∆V = ∆Q − T ∆S.
For a reversible process, ∆Q = T ∆S, and hence ∆G = 0. For an irreversible process,
∆Q < T ∆S, and hence ∆G < 0. Therefore, at constant temperature and pressure, the
Gibbs free energy tends to a minimum. This minimum is G(T, p) for the equilibrium state
at the given temperature and pressure.
9
II.
ONSAGER’S VARIATIONAL PRINCIPLE
A.
Reciprocal relations for linear irreversible thermodynamic processes
The heat flux J⃗ induced by temperature gradient ∇T is given by the constitutive equations
Ji = −
3
∑
λij ∇j T
(i = 1, 2, 3) ,
j=1
where λij are coefficients of heat conductivity. The heat conductivity tensor is symmetric
even in crystals of low symmetry (Stokes 1851).
B.
Onsager’s reciprocal symmetry derived from microscopic reversibility
For a closed system, consider the fluctuations of a set of (macroscopic) variables
αi (i = 1, ..., n) with respect to their most probable (equilibrium) values.
The entropy
of the system S has a maximum Se at equilibrium so that ∆S = S − Se can be written in
the quadratic form,
n
1∑
∆S (α1 , ..., αn ) = −
βij αi αj ,
2 i,j=1
where β is symmetric and positive definite. The probability density at αi (i = 1, ..., n) is
given by
f (α1 , ..., αn ) = f (0, ..., 0) e∆S/kB
where kB is the Boltzmann constant. The forces conjugate to αi are defined by
Xi =
n
∑
∂∆S
=−
βij αi
∂αi
j=1
which are linear combinations of αi not far from equilibrium.
Following the above definition of the forces, the equilibrium average (over the distribution
function f (α1 , ..., αn ) ) of αi Xj is given by
⟨αi Xj ⟩ = −kB δij .
Microscopic reversibility leads to the equality
⟨αi (t) αj (t + τ )⟩ = ⟨αj (t) αi (t + τ )⟩
10
for time correlation functions. In a certain domain not far from equilibrium, the macroscopic
variables αi (i = 1, ..., n) satisfy the linear equations
n
n
∑
∑
d
αi (t) = −
Mik αk (t) =
Lij Xj (t) .
dt
j=1
k=1
Here the Onsager kinetic coefficient matrix L is related to the rate coefficient matrix M via
the relation Lβ = M .
Onsager’s hypothesis is that fluctuations evolve in the mean according to the same macroscopic laws. Therefore, in evaluating the correlation function ⟨αi (t) αj (t + τ )⟩ for a short
time interval τ (a hydrodynamic time scale which is macroscopically short but microscopically long), αj (t + τ ) is given by
∑
d
αj (t + τ ) = αj (t) + τ αj (t) = αj (t) + τ
Ljk Xk (t) .
dt
k=1
n
It is worth pointing out that τ is macroscopically short for the linear expansion but microscopically long for the applicability of the macroscopic laws. It follows that ⟨αi (t) αj (t + τ )⟩
is given by
⟨αi (t) αj (t + τ )⟩ = ⟨αi (t) αj (t)⟩ + τ
n
∑
Ljk ⟨αi (t) Xk (t)⟩
k=1
= ⟨αi (t) αj (t)⟩ − τ kB Lji
Similarly, ⟨αj (t) αi (t + τ )⟩ is given by
⟨αj (t) αi (t + τ )⟩ = ⟨αj (t) αi (t)⟩ − τ kB Lij .
Comparing the above two time correlation functions, we obtain the reciprocal relations
Lij = Lji
from microscopic reversibility. Note that ⟨αi (t) αj (t)⟩ = ⟨αj (t) αi (t)⟩ by definition.
C.
Onsager’s variational principle based on the reciprocal symmetry
For small deviation from equilibrium, the system is in the linear response regime, where
the state variables αi (i = 1, ..., n) evolve according to the kinetic equations
α̇i = Lij Xj ,
11
(1)
or equivalently
Xi = Rij α̇j ,
(2)
where the kinetic coefficients Lij form a symmetric and positive definite matrix, and so do
the coefficients Rij , with Lij Rjk = δik . Off-diagonal entries Lij and Rij are referred to as
cross-coupling coefficients between different irreversible processes labeled by i and j. Under
the condition that the variables α are even, i.e., their signs remain invariant under time
reversal operation, Onsager derived the reciprocal relations
Lij = Lji ,
(3)
and consequently Rij = Rji , from the microscopic reversibility. It is worth emphasizing that
his derivation does not require detailed knowledge of the irreversible processes.
Based on Onsager’s reciprocal relations (3), the kinetic equations (2) can be used to formulate a variational principle governing the irreversible processes. This variational principle
states that for a closed system, the state evolution equations can be obtained by maximizing
the action function (or functional)
Ṡ (α, α̇) − ΦS (α̇, α̇)
(4)
with respect to the rates {α̇i }. Here Ṡ = Xi α̇i is the rate of change of the entropy, and the
dissipation function ΦS (α̇, α̇) = Rij α̇i α̇j /2 is half the rate of entropy production. For an
open system, however, there is an additional term Ṡ ∗ , which is the rate of entropy given by
the system to the environment, added to the action (4), leading to
O = Ṡ + Ṡ ∗ − ΦS (α̇, α̇) ,
(5)
which is called the Onsager-Machlup action. Note that Ṡ+Ṡ ∗ is still linear in {α̇i }. Onsager’s
variational principle states that for an open system, the state evolution equations can be
obtained by maximizing the Onsager-Machlup action O with respect to the rates {α̇i }. This
principle serves as a general framework for describing irreversible processes in the linear
response regime.
Onsager’s variational principle is an extension from Rayleigh’s principle of least energy
dissipation, and naturally it reduces to the latter for isothermal systems. In an isothermal
system, the rate of entropy given by the system to the environment can be expressed as
Ṡ ∗ = −Q̇/T = −U̇ /T , where T is the system temperature, Q̇ is the rate of heat transfer
12
from the environment to the system, and U̇ is the rate of change of the system energy, with
Q̇ = U̇ according to the first law of thermodynamics. Note that T is constant here. The
maximization of the Onsager-Machlup action (5) is equivalent to the minimization of the
so-called Rayleighian
R = Ḟ (α, α̇) + ΦF (α̇, α̇) ,
(6)
)
(
with respect to the rates {α̇i }. Here Ḟ ≡ U̇ −T Ṡ = −T Ṡ + Ṡ ∗ is the rate of change of the
Helmholtz free energy of the system, and the dissipation function ΦF (α̇, α̇) ≡ T ΦS (α̇, α̇)
is half the rate of free energy dissipation. As Ḟ is linear while ΦF is quadratic in the rates
{α̇i }, the principle of least energy dissipation leads to Ḟ = −2ΦF . For isothermal systems,
the Rayleighian can be written as
R=
∂F
1
α̇i + ζij α̇i α̇j ,
∂αi
2
(7)
where the first term in the right hand side is Ḟ and the second term is ΦF (α̇, α̇), which is in
a quadratic form with the friction coefficients ζij forming a symmetric and positive definite
matrix. Minimization of R with respect to the rates gives the kinetic equations
−
∂F
= ζij α̇j ,
∂αi
(8)
which can be interpreted as a balance between the reversible force −∂F/∂αi and the dissipative force linear in the rates.
It is worth emphasizing that although the variational principle is equivalent to the kinetic
equations combined with the reciprocal relations, the former possesses a notable advantage:
the variational form allows flexibility in the choice of state variables. Once these variables
are chosen, the conjugate forces are generated automatically via calculus of variations.
III.
A.
APPLICATIONS
Heat transport
Consider the heat transport in a crystal. The “forces” and “rates” (“velocities”) are
related by the constitutive equations
∑
1
− 2 ∇i T = Xi =
Rij Jj ,
T
j=1
3
13
where Xi (i = 1, 2, 3) are the forces and the components of the heat flux Jj (j = 1, 2, 3) are
the rates. Here the matrix R is the inverse of the Onsager coefficient matrix L, which is also
(
)
⃗ J⃗ is introduced in the form of
symmetric. The dissipation function ϕ J,
(
3
) 1∑
⃗
⃗
ϕ J, J ≡
Rij Ji Jj .
2 i,j=1
It is worth emphasizing that ϕ can be defined in this quadratic form because of the symmetry
in the matrices R and L. It is observed that substituting the constitutive equations into the
quadratic expression for ϕ yields
( )
3
3
3
) ∑
∑
∑
1
⃗ J⃗ ≡
2ϕ J,
Rij Ji Jj =
J i Xi =
J i ∇i
,
T
i,j=1
i=1
i=1
(
which equals the rate of entropy production per unit volume due to heat transport.
Let s denote the local entropy density in the system. Then, under the assumption of
local equilibrium, the rate of change of s is given by
)
ds
1(
=
−∇ · J⃗ ,
dt
T
where −∇ · J⃗ is the rate of local accumulation of heat. The rate of change of the total
entropy S is the volume integral
∫
Ṡ =
ds
dV =
dt
∫ (
)
1
⃗
− ∇ · J dV.
T
The rate of the entropy given off to the surrounding environment is given by the surface
integral
∗
∫ (
Ṡ =
Jn
T
)
dA,
where Jn is the outward normal component of the heat flux at the boundary. It follows that
)
∫ (
∫ ( )
1
Jn
∗
⃗
Ṡ + Ṡ =
− ∇ · J dV +
dA
T
T
( )
)
( )
∫
∫
∫ (
⃗
J
1
1
dV = J⃗ · ∇
dV.
=
− ∇ · J⃗ dV + ∇ ·
T
T
T
It can be shown that the constitutive equations for heat transport can be derived from
the variational principle
(
) [ ( )
]
⃗ J⃗ − Ṡ J⃗ + Ṡ ∗ (Jn ) = minimum,
Φ J,
14
⃗ are
where the temperature distribution is prescribed, and the rates, i.e., the heat flux J,
(
)
⃗ J⃗ is defined by
varied. Here Φ J,
∫
) ∫ (
)
⃗ J⃗ ≡ ϕ J,
⃗ J⃗ dV ≡
Φ J,
(
( )
Ṡ J⃗ is defined by
(
3
1∑
Rij Ji Jj
2 i,j=1
)
dV,
)
( ) ∫ ( 1
⃗
⃗
Ṡ J ≡
− ∇ · J dV,
T
and Ṡ ∗ (Jn ) is defined by
Ṡ
∗
(
) ∫ (J )
n
⃗
Jn ≡
dA,
T
with
( )
( )
( ) ∫
1
∗ ⃗
⃗
⃗
Ṡ J + Ṡ Jn ≡ J · ∇
dV.
T
(
) [ ( )
]
∗
⃗
⃗
⃗
The variation of Φ J, J − Ṡ J + Ṡ (Jn ) is given by
( )]
{ (
) [ ( )
]} ∫ ∑ [ ∂ (
)
∗
⃗ J⃗ − Ṡ J⃗ + Ṡ (Jn ) =
⃗ J⃗ − ∇k 1
δ Φ J,
ϕ J,
δJk dV,
∂J
T
k
k
from which we have
( )
∂ ( ⃗ ⃗)
1
ϕ J, J = ∇k
≡ Xk
∂Jk
T
according to the variational principle. We note that these are exactly the constitutive
equations
∑
j
( )
1
.
Rkj Jj = ∇k
T
As shown already, inserting the constitutive equations into the quadratic expression for ϕ
yields the equality
( )
) ∑
∂ ( ⃗ ⃗) ∑
1
⃗
⃗
2ϕ J, J =
Jk
ϕ J, J =
J k ∇k
,
∂J
T
k
k
k
(
and hence the integral form
(
)
( )
⃗ J⃗ = Ṡ J⃗ + Ṡ ∗ (Jn ) .
2Φ J,
( )
Note that Ṡ J⃗ + Ṡ ∗ (Jn ) is the rate of change of the entropy in the system and the
(
)
⃗ J⃗ is equal to
surrounding environment. Therefore, the rate of entropy production 2Φ J,
( )
the rate of change of the total entropy Ṡ J⃗ + Ṡ ∗ (Jn ) in an irreversible process governed
by the constitutive equations.
15
B.
Stokes equation and Navier slip boundary condition
The variational principle is now fully employed to investigate the Stokes flows with boundary slip. Below is a brief review, showing that the Stokes equation and the Navier slip
boundary condition can be derived from the principle of least energy dissipation.
Consider an incompressible Newtonian fluid in a region Ω with a solid boundary ∂Ω, and
neglect the inertial effect. The incompressibility condition reads ∇·⃗v = 0, and the boundary
is impermeable at which the normal velocity vn |∂Ω = 0. Here the rate is the velocity field
⃗v (⃗r) and the free energy is constant in time. The dissipation in the bulk region is due to
the viscosity η, and the corresponding dissipation function is
[
]
∫
1
2
Φv =
dV
η (∂i vj + ∂j vi ) .
4
Ω
(9)
If boundary slip occurs at the fluid-solid interface, then the corresponding dissipation function is given by
]
1 ( slip )2
,
Φs =
dS β ⃗v
2
∂Ω
∫
[
(10)
where β is the slip coefficient, and ⃗v slip is the slip velocity, defined as the tangential velocity
of the fluid relative to the solid at the fluid-solid interface. Here the solid boundary is still,
and hence ⃗v slip = ⃗v . The Rayleighian of the system is given by
R =Φv + Φs
] ∫
[
]
[
∫
1 2
1
2
η (∂i vj + ∂j vi ) +
dS β⃗v .
= dV
4
2
∂Ω
Ω
(11)
Combining the principle of least energy dissipation with the incompressibility condition,
[
]
∫
we have δ R − Ω dV π∂i vi = 0 for any ⃗v (⃗r) → ⃗v (⃗r) + δ⃗v (⃗r), with π being the Lagrange
multiplier. The Euler-Lagrange equations are the Stokes equation
[ (
)]
−∇π + ∇ · η ∇⃗v + ∇⃗v T = 0
(12)
in the bulk region, and the Navier boundary condition
n̂ · ↔
σ vis · ↔
τ + β⃗v slip = 0
(13)
(
)
at the solid boundary, where ↔
σ vis ≡ η ∇⃗v + ∇⃗v T is the viscous stress tensor, and ↔
τ ≡
↔
I − n̂n̂ with n̂ being the outward pointing (from fluid into solid) unit vector normal to ∂Ω.
↔
Note that the Lagrange multiplier π is the pressure. The total stress is ↔
σ ≡ −πI + ↔
σ vis .
16
C.
Lorentz reciprocal theorem
Prior to Onsager’s general work, there existed a few specific reciprocal relations studied
by Lord Kelvin and Helmholtz. In fluid dynamics, the hydrodynamic reciprocal relations,
known as the Lorentz reciprocal theorem, are also regarded as a special form of Onsager’s
reciprocal relations. Consider an incompressible Stokes flow in a region Ω with a solid
boundary ∂Ω. The velocity field ⃗v (⃗r) is governed by the Stokes equation (12), with the
no-slip boundary condition at ∂Ω. Suppose that in the same system, two velocity fields ⃗v (1)
and ⃗v (2) are both the solutions to Eq. (12), with their corresponding stress fields denoted by
↔(1)
σ
and ↔
σ (2) , respectively. The Lorentz reciprocal theorem states that
∫
∫
↔(1)
(2)
dS n̂ · σ · ⃗v =
dS n̂ · ↔
σ (2) · ⃗v (1) ,
(14)
∂Ω
∂Ω
where n̂ is the outward pointing (from fluid into solid) unit vector normal to ∂Ω. The proof
is as follows. The left hand side of Eq. (14) can be expressed as
∫
∫
( (1) (2) )
↔(1)
(2)
dS n̂ · σ · ⃗v = dV ∇ · ↔
σ · ⃗v
∂Ω
Ω
∫
(
)
= dV ∇ · ↔
σ (1) · ⃗v (2) + ↔
σ (1) : ∇⃗v (2)
∫Ω
[
)(
)]
η ( (1)
(2)
(1)
(2)
(2)
(1)
= dV −p δij ∂i vj +
∂i vj + ∂j vi
∂i vj + ∂j vi
2
∫Ω
)(
)
η ( (1)
(1)
(2)
(2)
∂i vj + ∂j vi
∂i vj + ∂j vi
,
= dV
2
Ω
(15)
where the Stokes equation, ∇ ·⃗v = 0, and the symmetry of ↔
σ are used. It is readily seen that
the right hand side of of Eq. (14) leads to the same expression. Furthermore, the Lorentz
reciprocal theorem (14) can be expressed as
(1) (2)
(2) (1)
Fk ẋk = Fk ẋk ,
(16)
where ẋk are the generalized velocities of the solid objects and Fk are the generalized dissipative forces conjugate to ẋk . Note that the no-slip boundary condition is applied to move
from ⃗v of the fluid to ẋk of the solid. Due to the linearity of the Stokes flows, we have the
linear dependence of the forces on the rates:
Fk = ζkl ẋl ,
17
(17)
where the friction coefficients ζkl form a positive definite matrix. It follows from Eq. (16)
that the Lorentz reciprocal theorem can be expressed as
ζkl = ζlk ,
(18)
meaning that the matrix formed by the friction coefficients ζkl is symmetric.
In the above discussion, the Lorentz reciprocal theorem expressed in Eqs. (16) and (18)
is derived from the Stokes equation and the no-slip boundary condition. We have already
shown that the Stokes equation (12) and the Navier boundary condition (13) can be simultaneously obtained from the principle of least energy dissipation. It is therefore expected
that the hydrodynamic reciprocal relations can be generalized to describe the Stokes flows
with the Navier boundary condition.
Consider the same system with the Navier boundary condition. The velocity of the solid
⃗ . From the Navier boundary condition (13), we readily obtain
boundary ∂Ω is denoted by W
∫
∫
↔(1)
slip(2)
dS n̂ · σ · ⃗v
=
dS n̂ · ↔
σ (2) · ⃗v slip(1) .
(19)
∂Ω
∂Ω
⃗ = ⃗v − ⃗v slip on ∂Ω. By combining Eqs. (14)
Meanwhile, Eq. (14) still holds. Note that W
and (19), we obtain the generalized form of the hydrodynamic reciprocal relations
∫
∫
↔(1) ⃗ (2)
⃗ (1) .
dS n̂ · σ · W =
dS n̂ · ↔
σ (2) · W
∂Ω
(20)
∂Ω
⃗ = ⃗v on ∂Ω. With the Lorentz
Note that the no-slip limit is obtained as β → ∞ with W
reciprocal theorem generalized from Eq. (14) to (20), it can be further expressed as Eq. (16),
which results in the symmetry in Eq. (18). We emphasize that in the presence of boundary
slip, we need Eq. (20) in order to arrive at Eqs. (16) and (18). It is remarkable that the
reciprocal symmetry is preserved in the Stokes flows with the Navier slip condition.
To use Eq. (18) for the present study, we consider the solid boundary ∂Ω consisting of the
surfaces of N rigid bodies ∂Ωi (i = 1, ..., N ), each in a motion described by the translational
velocity V⃗ i and the angular velocity ⃗ω i . The solid velocity at ⃗r on ∂Ωi can be expressed as
⃗ (⃗r) = V⃗ i + ⃗ω i × δ⃗ri
W
(21)
where δ⃗ri is measured relative to the center of mass of the i-th rigid body. Then we have
∫
⃗ (2)
dS n̂ · ↔
σ (1) · W
=
∂Ω
N
∑ (∫
i=1
∂Ωi
↔(1)
dS n̂ · σ
)
· V⃗
i(2)
+
N [∫
∑
i=1
18
∂Ωi
(
↔(1) )
dSδ⃗r × n̂ · σ
i
]
·ω
⃗ i(2) ,
where
∫
∂Ωi
↔
dS n̂·σ
is the total force by the i-th rigid body on the fluid, and
∫
∂Ωi
dSδ⃗ri ×(n̂ · ↔
σ)
is the total torque by the i-th rigid body on the fluid. This leads to a specific form of Eq. (16),
in which V⃗ i and ⃗ω i (i = 1, ..., N ) are the generalized velocities of the rigid bodies ẋk , and
∫
∫
↔
dS
n̂
·
σ
and
dSδ⃗ri × (n̂ · ↔
σ ) are their conjugate generalized forces Fk .
i
∂Ω
∂Ωi
Finally we emphasize that the Lorentz reciprocal theorem is valid only when the slip
length ls = η/β is a material constant, which makes the Navier boundary condition linear.
D.
Nematic liquid crystals
The Leslie-Ericksen hydrodynamic theory for nematic liquid crystals gives the dissipative
stress tensor ↔
σ and the torque density by the director on the fluid ⃗Γ as
(
( ↔)
( ↔)
↔)
↔
⃗ + α3 N⃗
⃗ n + α4 A + α5⃗n ⃗n · A + α6 ⃗n · A ⃗n,
σ =α1 ⃗n⃗n : A ⃗n⃗n + α2⃗nN
↔
(22)
(
)
↔
⃗Γ = ⃗n × γ1 N
⃗ + γ2 A · ⃗n ,
(23)
(
)
↔
where αi (i = 1, ..., 6) are phenomenological parameters, ⃗n is the director, A ≡ 12 ∇⃗v + ∇⃗v T
⃗ ≡ ⃗n˙ − ⃗ν × ⃗n = (⃗ω − ⃗ν ) × ⃗n is the velocity of the
is the rate-of-strain tensor of flow, and N
director relative to the fluid, with ω
⃗ being the angular velocity of the director and ⃗ν ≡ 21 ∇×⃗v
being the angular velocity of the fluid. In addition, γ1 and γ2 are given by γ1 = α3 − α2 and
γ2 = α6 − α5 . Only five of the six αi (i = 1, ..., 6) parameters are independent because of
the Parodi relation
α2 + α3 = α6 − α5 ,
(24)
which results from the Onsager reciprocal relations.
E.
Cross coupling in gas flows in micro-channels
In rarefied gases, mass and heat transport processes interfere with each other, leading to
the mechano-caloric effect and thermo-osmotic effect, which are of interest to both theoretical
study and practical applications. We employ the unified gas-kinetic scheme to investigate
these cross coupling effects in gas flows in micro-channels. Our numerical simulations cover
channels of planar surfaces and also channels of ratchet surfaces, with Onsager’s reciprocal
relation verified for both cases. For channels of planar surfaces, simulations are performed
in a wide range of Knudsen number and our numerical results show good agreement with
19
the literature results. For channels of ratchet surfaces, simulations are performed for both
the slip and transition regimes and our numerical results not only confirm the theoretical
prediction [Phys. Rev. Lett. 107, 164502 (2011)] for Knudsen number in the slip regime but
also show that the off-diagonal kinetic coefficients for cross coupling effects are maximized
at a Knudsen number in the transition regime. Finally, a preliminary optimization study is
carried out for the geometry of Knudsen pump based on channels of ratchet surfaces.
In a closed system out of equilibrium, the rate of entropy production can be expressed as
dS ∑
=
Ji Xi ,
dt
i=1
N
(25)
where S is the entropy, Ji are the thermodynamic fluxes, and Xi are the conjugate thermodynamic forces. For small deviation away from equilibrium, we have the linear relations
between Ji and Xi :
Ji =
N
∑
Lij Xj ,
(26)
j=1
where Lij are the kinetic coefficients. Onsager’s reciprocal relations state that Lij and Lji
are equal as a result of microscopic reversibility. Starting from the Gibbs equation, the
thermodynamic fluxes and forces can be identified for gas flows, and the corresponding
constitutive equations can be derived.
A schematic illustration of the cross coupling in a channel of planar surfaces can be
found in figure 2, where a long channel is confined by two parallel solid plates separated by
a distance H and connected with two reservoirs. The left reservoir is maintained at pressure
p0 − ∆p/2 and temperature T0 − ∆T /2 while the right reservoir is maintained at p0 + ∆p/2
and T0 + ∆T /2. We use ∆p < 0 and ∆T > 0 in our simulations, with |∆p/p0 | ≪ 1 and
|∆T /T0 | ≪ 1 to ensure the linear response. Usually, a mass flux to the right is generated
by the pressure gradient due to ∆p < 0 and a heat flux to the left is generated by the
temperature gradient due to ∆T > 0. For rarefied gas, however, ∆p also contributes to the
heat flux and ∆T also contributes to the mass flux. These cross coupling effects are called
the mechano-caloric effect and thermo-osmotic effect respectively.
For a single-component gas, the rate of entropy production can be expressed as
( )
( ν)
dS
1
= JM ∆ −
+ JE ∆
,
dt
T
T
(27)
where ν is the chemical potential per unit mass, JE and JM are the energy flux and mass
flux from the left reservoir to the right reservoir, and ∆ means the quantity on the right
20
minus the quantity on the left. Here ν and JE can be written as
ν = h − T s,
(28)
JE = JQ + hJM ,
(29)
where s and h are the entropy and enthalpy per unit mass, and JQ is the heat flux. Together
with the Gibbs-Duhem equation
dν = −sdT + dp/ρ,
(30)
where ρ is the mass density. Equation (27) becomes
dS
1
1
= − JM ∆p − 2 JQ ∆T.
dt
ρT
T
(31)
According to equation (31), the thermodynamic forces and fluxes are connected in the
form of


  
−1 −1
L
L
−ρ T ∆p
J
,
 M  =  M M M Q  0 0
LQM LQQ
−T0−2 ∆T
JQ
(32)
with
LM Q = LQM ,
(33)
due to Onsager’s reciprocal relations. The detailed mechanism may vary with geometric
configuration and rarefaction. Here and throughout the paper, the subscript ‘0’ denotes the
reference state from which various deviations (in pressure, temperature, etc) are measured.
In the free molecular regime and with specular reflection on plates, the gas molecules
travel ballistically from on side to the other and the distribution function at any point can
be treated as a combination of two half-space Maxwellians from the two reservoirs. The
kinetic coefficients in equation (32) can be analytically derived in this case, given by


LM M LM Q


LQM LQQ


√
ρ
/p
−1/2
0
0
Hρ0 T0 8kB T0 
.
=
4
πm
−1/2 9p /4ρ
0
(34)
0
where kB is the Boltzmann constant and m is the molecular mass.
If the temperature gradient is imposed on the plates and the gas molecules are diffusely
reflected, then the mass flux due to the temperature gradient is generated by thermal creep
21
on the plates. The kinetic coefficients in this case have been calculated by several authors
using different methods. Assuming the length to height ratio of the channel is fixed and
noting ρλ = constant and µ ∝ ρ0 T 1/2 for hard-sphere molecules, the average velocity U
induced by thermal creep can be estimated from the Maxwell slip boundary condition,
U∼
µ0
∆T
∇T ∝ Kn √ ,
ρ0 T0
T0
(35)
where λ is the mean free path, µ is the dynamic viscosity independent of the density, and
Kn = λ0 /H is the Knudsen number. In later sections, we will show that LM Q and LQM are
equal and increase with the increasing Kn.
F.
1.
Cross coupling in a mixture of fluids
Ideal fluids
The continuity equation is given by
∂
ρ + ∇ · (ρ⃗v ) = 0
∂t
where ρ is the mass density. The momentum equation is given by
∂
d
(ρ⃗v ) + ∇ · (ρ⃗v⃗v ) = ρ ⃗v = −∇p
∂t
dt
where
d
⃗v
dt
≡
∂
⃗v
∂t
+ ⃗v · ∇⃗v is the rate of change of the velocity of a given fluid particle, and
p is the pressure. This is Euler’s equation and is one of the fundamental equations of fluid
dynamics. The motion of an ideal fluid is adiabatic and in adiabatic motion the entropy
of any fluid particle remains constant as the particle moves about in space. Therefore, the
entropy equation is given by
∂
(ρs) + ∇ · (ρs⃗v ) = 0
∂t
where s is the entropy per unit mass.
Now we are ready to derive the energy equation. The kinetic energy density is 12 ρ⃗v 2 and
the internal energy density is ρε where ε is the internal energy per unit mass. Using the
continuity equation and the momentum equation, we have
)
[(
) ]
(
∂ 1 2
1 2
ρ⃗v = −⃗v · ∇p − ∇ ·
ρ⃗v ⃗v
∂t 2
2
22
for the kinetic energy. As to the internal energy, we have d (ρε) = ρT ds + hdρ where
h = ε + p/ρ is the enthalpy per unit mass. It follows that for an ideal fluid we have
∂
∂
∂
(ρε) = ρT s + h ρ = −ρT⃗v · ∇s − h∇ · (ρ⃗v )
∂t
∂t
∂t
by use of the entropy equation and the continuity equation. Putting the two energies together, we have
) ]
(
)
[(
∂ 1 2
1 2
ρ⃗v + ρε = −⃗v · ∇p − ∇ ·
ρ⃗v ⃗v − ρT⃗v · ∇s − h∇ · (ρ⃗v ) .
∂t 2
2
To proceed, we note that from thermodynamics we have dh = T ds + (1/ρ)dp, ρdh = ρT ds +
(
)
∂ 1
dp, and hence ρ∇h = ρT ∇s + ∇p. It follows that ∂t
ρ⃗v 2 + ρε can be expressed as
2
(
)
[(
) ]
∂ 1 2
1 2
ρ⃗v + ρε = −∇ ·
ρ⃗v + ρh ⃗v ,
∂t 2
2
(
)
which shows that the energy flux is given by 12 ρ⃗v 2 + ρh ⃗v . Note that we have ρh⃗v rather
than ρε⃗v in the energy flux because the work done by pressure forces is to be included.
2.
Viscous fluids
Now we turn to viscous fluids. The continuity equation remains unchanged and the
momentum equation is given by
∂
d
↔′
(ρ⃗v ) + ∇ · (ρ⃗v⃗v ) = ρ ⃗v = −∇p + ∇ · σ
∂t
dt
↔′
where σ is the viscous stress tensor. The energy equation is given by
(
)
[(
) ]
(↔′ )
∂ 1 2
1 2
ρ⃗v + ρε = −∇ ·
ρ⃗v + ρh ⃗v + ∇ · σ · ⃗v − ∇ · ⃗q
∂t 2
2
where ⃗q is the heat flux. Note that the energy equation for viscous fluids takes into account
the work done by viscous forces and the thermal conduction, which are absent in ideal fluids.
The entropy equation can be derived by use of the above equations and thermodynamic
relations. The derivation is straightforward and the procedure can be outlined as follows.
Using the continuity equation and the momentum equation, we can obtain an equation for
)
)
(
(
∂ 1
∂ 1
ρ⃗v 2 . Combining the energy equation and the equation for ∂t
ρ⃗v 2 , we can obtain an
∂t 2
2
equation for
∂
∂t
(ρε):
∂
↔′
(ρε) = −∇ · (ρh⃗v ) + σ : ∇⃗v − ∇ · ⃗q + ⃗v · ∇p.
∂t
23
Combining the above equation with
ρT
∂
∂t
∂
∂
(ρε) = ρT ∂t
s + h ∂t
ρ, we obtain
∂
↔′
s = σ : ∇⃗v − ∇ · ⃗q + ⃗v · ∇p − ρ⃗v · ∇h,
∂t
which becomes
(
ρT
∂
s + ⃗v · ∇s
∂t
)
↔′
= σ : ∇⃗v − ∇ · ⃗q,
with the help of −ρ∇h + ∇p = −ρT ∇s. Finally we obtain the entropy equation
( )
∂
1 ↔′
1
⃗q
+ σ : ∇⃗v + ⃗q · ∇ ,
(ρs) + ∇ · (ρs⃗v ) = −∇ ·
∂t
T
T
T
where
q⃗
T
is the entropy flux,
1 ↔′
σ
T
: ∇⃗v is the rate of entropy production due to viscous
dissipation, and ⃗q · ∇ T1 is the rate of entropy production due to thermal conduction.
3.
A mixture of fluids
Let us start from the continuity equations of two miscible components, with one labeled
by the subscript “1” and the other labeled by the subscript “2”. They read
∂
ρ1 + ∇ · (ρ1⃗v1 ) = 0
∂t
and
∂
ρ2 + ∇ · (ρ2⃗v2 ) = 0,
∂t
in which ⃗vi (i = 1, 2) is the velocity of a particular species. The mass density ρ and velocity
⃗v of the mixture are defined by ρ = ρ1 + ρ2 and ρ⃗v = ρ1⃗v1 + ρ2⃗v2 . Physically, ⃗v is the massaveraged velocity which is a field variable that enters into the hydrodynamic momentum
equation. The local relative concentration c is defined by ρc = ρ1 − ρ2 . It follows that ρc⃗v
equals (ρ1 − ρ2 )⃗v .
Adding the continuity equations for the two components gives the continuity equation
∂
ρ + ∇ · (ρ⃗v ) = 0
∂t
for ρ. The diffusive flux ⃗j is defined through the equation ρ1⃗v1 − ρ2⃗v2 = ρc⃗v + ⃗j. It follows
that ⃗j is given by ⃗j = ρ1 (⃗v1 − ⃗v ) − ρ2 (⃗v2 − ⃗v ). Note that ρ1 (⃗v1 − ⃗v ) + ρ2 (⃗v2 − ⃗v ) = 0 —
the diffusion discussed here is defined relative to the motion of the center of mass of a fluid
24
element. Rewriting the difference between the continuity equations for the two components
as
(
)
∂
(ρc) + ∇ · ρc⃗v + ⃗j = 0,
∂t
we obtain
)
∂
d
c + ⃗v · ∇c = ρ c = −∇ · ⃗j.
ρ
∂t
dt
By introducing µ as an appropriately defined chemical potential of the mixture, we have
(
thermodynamic equations
d (ρε) = ρT ds + hdρ + ρµdc
and
ρdh = ρT ds + dp + ρµdc
for ε and h. They will be used when we derive the entropy equation from the energy equation.
To proceed, we employ the momentum equation
∂
d
↔′
(ρ⃗v ) + ∇ · (ρ⃗v⃗v ) = ρ ⃗v = −∇p + ∇ · σ
∂t
dt
and the energy equation
(
)
[(
) ]
(↔′ )
∂ 1 2
1 2
ρ⃗v + ρε = −∇ ·
ρ⃗v + ρh ⃗v + ∇ · σ · ⃗v − ∇ · ⃗q,
∂t 2
2
which look the same as those used for viscous fluids of one component. We would like to
↔′
point out that while σ is still the viscous stress tensor, the physical meaning of ⃗q is not clear
at the moment — it is expected to represent the total energy flux due to thermal conduction
and diffusion. This is to be clarified by the explicit form of the entropy equation.
In order to derive the entropy equation, we first combine the continuity equation and
(
)
∂ 1
2
ρ⃗
v
. We then combine the energy
the momentum equation to obtain an equation for ∂t
2
(
)
∂ 1
∂
equation and the equation for ∂t
ρ⃗v 2 to obtain an equation for ∂t
(ρε):
2
∂
↔′
(ρε) = −∇ · (ρh⃗v ) + σ : ∇⃗v − ∇ · ⃗q + ⃗v · ∇p.
∂t
Combining the above equation with
∂
∂t
∂
∂
∂
(ρε) = ρT ∂t
s + h ∂t
ρ + ρµ ∂t
c from thermodynamics,
we obtain
ρT
which becomes
(
ρT
∂
∂
↔′
s = σ : ∇⃗v − ∇ · ⃗q + ⃗v · ∇p − ρ⃗v · ∇h − ρµ c,
∂t
∂t
)
(
)
∂
∂
↔′
s + ⃗v · ∇s = σ : ∇⃗v − ∇ · ⃗q − ρµ
c + ⃗v · ∇c ,
∂t
∂t
25
(∂
)
with the help of −ρ∇h + ∇p = −ρT ∇s − ρµ∇c. Using ρ ∂t
c + ⃗v · ∇c = −∇ · ⃗j, we have
)
(
(
)
∂
↔′
↔′
s + ⃗v · ∇s = σ : ∇⃗v − ∇ · ⃗q + µ∇ · ⃗j = σ : ∇⃗v − ∇ · ⃗q − µ⃗j − ⃗j · ∇µ,
ρT
∂t
which gives the entropy equation
∂
(ρs) + ∇ · (ρs⃗v ) = −∇ ·
∂t
in which
q −µ⃗j
⃗
T
(
⃗q − µ⃗j
T
)
+ π,
is the entropy flux and π is the rate of entropy production per unit volume,
given by
π=
(
)
1 ↔′
1
∇µ
σ : ∇⃗v + ⃗q − µ⃗j · ∇ − ⃗j ·
.
T
T
T
(
)
1 ↔′
σ : ∇⃗v is the rate of entropy production due to viscous dissipation, and ⃗q − µ⃗j ·
T
is the rate of entropy production due to thermal conduction and diffusion. The
∇ T1 − ⃗j · ∇µ
T
Here
latter can be written as
(
)
(
) ( ∇T )
∇µ
⃗
⃗
⃗q − µj · − 2 + j · −
T
T
(
)
in which ⃗q − µ⃗j and ⃗j are the fluxes associated with thermal conduction and diffusion, and
− ∇T
and − ∇µ
are the conjugate forces. Now we are ready to write down the constitutive
T2
T
equations
(
)
(
)
∇µ
∇T
2
− βT
T
T2
(
)
(
)
∇µ
∇T
2
⃗
⃗q − µj = −δT
− γT
T
T2
⃗j = −αT
with the reciprocal relation βT 2 = δT for the cross coupling between thermal conduction
and diffusion.
FIGURES
26
1
2
FIG. 1. A closed system made up of two interconnected subsystems.
p0 − ∆p/2
T0 − ∆T /2
mass flux JM
H
heat flux JQ
p0 + ∆p/2 (∆p < 0)
T0 + ∆T /2 (∆T > 0)
L
FIG. 2. A schematic illustration of the cross coupling in a channel of planar surfaces.
27